Theorem pmtrsn 19550

Description: The value of the transposition generator function for a singleton is empty, i.e. there is no transposition for a singleton. This also holds for 𝐴 ∉ V, i.e. for the empty set {𝐴} = ∅ resulting in (pmTrsp‘∅) = ∅. (Contributed by AV, 6-Aug-2019.)

Assertion

Ref	Expression
pmtrsn	⊢ (pmTrsp‘{𝐴}) = ∅

Proof of Theorem pmtrsn

Dummy variables 𝑝 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5393	. . 3 ⊢ {𝐴} ∈ V
2		eqid 2761	. . . 4 ⊢ (pmTrsp‘{𝐴}) = (pmTrsp‘{𝐴})
3	2	pmtrfval 19481	. . 3 ⊢ ({𝐴} ∈ V → (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))))
4	1, 3	ax-mp 5	. 2 ⊢ (pmTrsp‘{𝐴}) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
5		eqid 2761	. . . . 5 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
6	5	dmmpt 6222	. . . 4 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
7		2on0 8446	. . . . . . . . 9 ⊢ 2o ≠ ∅
8		ensymb 8977	. . . . . . . . . 10 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅)
9		en0 8993	. . . . . . . . . 10 ⊢ (2o ≈ ∅ ↔ 2o = ∅)
10	8, 9	bitri 277	. . . . . . . . 9 ⊢ (∅ ≈ 2o ↔ 2o = ∅)
11	7, 10	nemtbir 3052	. . . . . . . 8 ⊢ ¬ ∅ ≈ 2o
12		snnen2o 9183	. . . . . . . 8 ⊢ ¬ {𝐴} ≈ 2o
13		0ex 5254	. . . . . . . . 9 ⊢ ∅ ∈ V
14		breq1 5100	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → (𝑦 ≈ 2o ↔ ∅ ≈ 2o))
15	14	notbid 320	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (¬ 𝑦 ≈ 2o ↔ ¬ ∅ ≈ 2o))
16		breq1 5100	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝑦 ≈ 2o ↔ {𝐴} ≈ 2o))
17	16	notbid 320	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → (¬ 𝑦 ≈ 2o ↔ ¬ {𝐴} ≈ 2o))
18	13, 1, 15, 17	ralpr 4656	. . . . . . . 8 ⊢ (∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o ↔ (¬ ∅ ≈ 2o ∧ ¬ {𝐴} ≈ 2o))
19	11, 12, 18	mpbir2an 721	. . . . . . 7 ⊢ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o
20		pwsn 4855	. . . . . . . 8 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
21	20	raleqi 3317	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o ↔ ∀𝑦 ∈ {∅, {𝐴}} ¬ 𝑦 ≈ 2o)
22	19, 21	mpbir 233	. . . . . 6 ⊢ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o
23		rabeq0 4339	. . . . . 6 ⊢ ({𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅ ↔ ∀𝑦 ∈ 𝒫 {𝐴} ¬ 𝑦 ≈ 2o)
24	22, 23	mpbir 233	. . . . 5 ⊢ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} = ∅
25	24	rabeqi 3426	. . . 4 ⊢ {𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V}
26		rab0 4336	. . . 4 ⊢ {𝑝 ∈ ∅ ∣ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) ∈ V} = ∅
27	6, 25, 26	3eqtri 2788	. . 3 ⊢ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
28		mptrel 5794	. . . 4 ⊢ Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))
29		reldm0 5900	. . . 4 ⊢ (Rel (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) → ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅))
30	28, 29	ax-mp 5	. . 3 ⊢ ((𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅ ↔ dom (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅)
31	27, 30	mpbir 233	. 2 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 {𝐴} ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ {𝐴} ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = ∅
32	4, 31	eqtri 2784	1 ⊢ (pmTrsp‘{𝐴}) = ∅

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453   ∖ cdif 3899  ∅c0 4283  ifcif 4477  𝒫 cpw 4552  {csn 4579  {cpr 4581  ∪ cuni 4862   class class class wbr 5097   ↦ cmpt 5178  dom cdm 5643  Rel wrel 5648  ‘cfv 6516  2_oc2o 8425   ≈ cen 8918  pmTrspcpmtr 19472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-1o 8431  df-2o 8432  df-er 8672  df-en 8922  df-pmtr 19473

This theorem is referenced by:  psgnsn  19551